Ask Dr. Time: Explaining Mathematics

Today’s question is surprisingly tricky, as even the letter writer acknowledges:

My question is one I’m fumbling to articulate. I’m a math teacher and writer. (I’m a writer in the sense that I write, not in the sense that I get published or paid for writing.) I write a lot about teaching, but I’ve also been trying to get a handle on how I can write about math.

Here’s the question: is it possible to write about math in a deep and accessible way?

This is a question that sends me off on a lot of different questions. What does it mean to understand math? What does it mean to understand a metaphor? Are there are great literary works that are also mathematical?

Ultimately, though, I don’t know how to think about this yet. I’m hoping to eventually figure this out by learning math and writing about it…but that’s slow, so maybe Dr. Time can offer advice?

The obvious answer to this question is yes, of course it’s possible to write about math in a deep and accessible way. Bertrand Russell won a Nobel Prize in Literature. Godel, Escher, Bach is a 777-page doorstop that’s also a beloved bestseller. If you’re looking to satisfy an existence requirement, that book has your back. I’ll even stipulate that for every intellectual subject, not just mathematics, there exists a work that satisfies this deep-but-accessible requirement. It’s just like how there’s always a bigger prime number. It’s out there; we just have to find it.

On the other hand, math seems hard. And I think it seems hard for Reasons. Here’s a big one: mathematicians and popularizers of mathematics are perhaps understandably obsessed with understanding mathematics as such. The want to explain the totality of mathematics, or the essence, rather than finer problems like distinguishing between totalities and essences.

If you look at the other sciences, they don’t do this. It’s only very rarely that you get a Newton, Darwin, or Einstein who sets out to grab his or her entire subject with both hands and rethink our fundamental understanding of its foundations. Imagine a biologist who wants to explain life, in its essence and totality, at the micro and macro level. They’d be understandably stumped. Even physicists, when they want to explain something big and weird to the public, stick to things like a subatomic particle they’re hoping to discover or the behavior of one of Saturn’s moons. They don’t try to explain physics. They explain a problem in physics.

When mathematicians do that, they’re usually pretty successful. The Konigsberg Bridge Problem is charming as hell. Russell’s and Godel’s paradoxes have whole books written about them, but can also be told in the form of jokes. Even Fourier Transforms can be broken down and made beautiful with a little bit of technical help.

So I think the key, in part, is to resist that mathematicians’ tendency to abstract away individual problems into general solutions or categories of solutions or entire subfields, and spend some time with the specific problems that mathematicians are or have been interested in. But it also helps a lot if, in that specific problem, you get that mathematical move of discarding whatever doesn’t matter to the structure of the problem. After all, that’s a big part of what you’re trying to teach: how to think like a mathematician. You just to have to unlearn what a mathematician already assumes first.